Idempotent measures on locally compact semigroups

Abstract

A conjecture that the support of an /■"'-invariant regular finite measure on a locally compact semigroup is a left group is proven. Moreover we also prove that the support of an idempotent measure on a locally compact semigroup is completely simple, thus extending a well-known result of Pym and Heble-Rosenblatt on compact semigroups to the locally compact case. These results are also shown to be true in a complete metric semi-group.

Kawada-Itô-Kelley theorem for quantum semigroups

“…We could not prove that in this case p ⊥ ω is compact which then would imply that ω is of Haar type. However if S is a classical semigroup this follows from [21] as explained in Section 1.…”

“…Theory of idempotent measures on a locally compact semigroups was developed in a number of papers, see e.g. [26], [32] and culminates in [21]. The latter contains the following structural characterization of idempotent measures: let S be a locally compact semigroup and ω ∈ C 0 (S) * an idempotent state with the support S ω ⊂ S. Then S ω is a closed subsemigroup and there are locally compact spaces X, Y equipped with Borel probability measures µ X and µ Y , a compact group G, a map φ : Y × X → G such that:…”

On r*-Invariant Measure on a Locally Compact Semigroup with Recurrence

“…A regular Borel measure n is said to be r*-invariant on a locally compact semigroup if ii(Ba~1) = n(B) for all Borel sets B and points a of S, where BaT 1 ={xeS, xa eB}. In [1] Argabright conjectured that the support of an r*-invariant measure on a locally compact semigroup is a left group, Mukherjea and Tserpes [4] proved this conjecture in the case that the measure is finite; however their method of proof fails when the measure is infinite. In this paper the following theorem is proved:…”

“…Since F is a right ideal in S, then aF is a right ideal and a semigroup in S. The measure J LL induces a r*-invariant measure on aF which is the restriction of the measure / UL to aF. Since Tserpes and Mukherjea [4] proved the conjecture when fx is finite, we may assume that the induced measure is infinite on aF, which is left cancellative [4]. In view of [4] and [6], F is now assumed to be left cancellative.…”